Have a neighbor that is larger than most and has problems with spokes breaking. How much difference is there between heavy duty spokes and plain ones? How much weight increase can be expected going to heavy spokes?

If your neighbor is "larger than most" and having problems with spoke breakage, the additional weight of heavier-gauge spokes is irrelevant.

Spoke breakage is usually not the result of the style or size of spokes, but the result of an incorrectly built or tensioned wheel. Lighter spokes, especially 'butted' spokes can actually be built into a stronger wheel because the added flex through the centre of the spoke takes some of the stress away from the elbows (where breakage normally occurs).

I have been up close to 300 lbs in the past and use standard 14 ga. spokes and do not have a problem with spoke breakage.

DCBO is right on!
Spoke failure, aside from an accident, is most often caused by metal fatigue in the spoke which in turn is caused by low spoke tension not the strength or guage of the spoke itself.
A wheel built to last will do so even with a heavy rider.

+1, I've built wheels using 14g butted spokes for tandems that have gone on loaded cross country tours without problems. There's plenty of strength and resiliency in today's spokes and rims. Proof of that is the tandem wheels, and at the other end, low spoke count wheels using similar spokes.

But the problem often has nothing to do with the wheel, but rather with the rider. Riding habits can be at least as important as wheel quality. I've known gorillas (figurative) who never had wheel issues, and a ballerina (literally) who was death on her wheels.

If you want to build super duty wheels for your friend you might use 13/14g single butted spokes, or one of the triple butted spokes, either of which will add 20% more strength at the critical elbow stress area. But I suspect that to really solve the problem, you'll need to look at your friends riding habits. Things like standing and allowing the knees to absorb loads over bumps, not excessively rocking the bike in hard climbs, and smoother pedaling cadence can do wonders in preventing wheel problems.

How many spokes in the wheels you are talking about. For heavy friends I build with 36 double butted spokes and they do fine.